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How to evaluate one's behavior toward 'bad' individuals? Exploring good social norms in promoting cooperation in spatial public goods games high-reputation threshold environments, a well-mixed population structure can even promote cooperation more significantly than a lattice network. However, increasing reward intensity for cooperating with 'bad' individuals cannot further improve cooperation, but in a high-reputation threshold environment, increasing punishment intensity for defecting 'bad' individuals can further improve cooperation. This research extends the use of statistical physics to study the evolution of cooperation from the perspective of reputation-based dynamics.

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The spatial public goods game (SPGG) is considered as a basic model for understanding the emergence of cooperation under the network reciprocity mechanism. For an intra-group game in SPGG, it is generally assumed that the total investment of all investors is linearly amplified and distributed within the group. In real situations, the value of public goods may be manifested as a nonlinear relationship of the total investment. In addition, the nonlinear function relationships are also heterogeneous for dierent public goods. Inspired by this observation, we made two improvements in the traditional SPGG. First, we introduced a scale return coecient in the group of PGG to describe the nonlinear relationship between the value of public goods and the amount of investment. Then, we considered the heterogeneity of the scale return coecient between dierent groups in SPGG. We investigated three types of distributions including uniform distribution, exponential distribution, and power-law distribution. Besides, we introduced an amplitude parameter to measure the degree of heterogeneity of the scale return coecient. Simulation experiments on the square lattice network demonstrate that the larger the amplitude parameter for the same kind of distribution, which means that the stronger the heterogeneity, the more conducive to the cooperation of the population. Under

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